Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting, derived, Sylow | #62895 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | C22⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | #62895 | C3 |
| LCS | G ▷ #62895 | |
| UCS | C1 ◁ C22 | |
1449 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1449 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|c^2=h,d^2=h,e^2=h,b^a=bce,c^a=dg,d^a=cdfghi,e^a=bdghi,f^a=fgi,g^a=fhi,c^b=ch,d^b=di,e^b=eh,g^b=gi,d^c=dh,f^e=fi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085139);